Volumetric heat capacity

Last updated January 15, 2025

The volumetric heat capacity of a material is the heat capacity of a sample of the substance divided by the volume of the sample. It is the amount of energy that must be added, in the form of heat, to one unit of volume of the material in order to cause an increase of one unit in its temperature. The SI unit of volumetric heat capacity is joule per kelvin per cubic meter, J⋅K⁻¹⋅m⁻³.

This quantity may be convenient for materials that are commonly measured by volume rather than mass, as is often the case in engineering and other technical disciplines. The volumetric heat capacity often varies with temperature, and is different for each state of matter. While the substance is undergoing a phase transition, such as melting or boiling, its volumetric heat capacity is technically infinite, because the heat goes into changing its state rather than raising its temperature.

The volumetric heat capacity of a substance, especially a gas, may be significantly higher when it is allowed to expand as it is heated (volumetric heat capacity at constant pressure) than when is heated in a closed vessel that prevents expansion (volumetric heat capacity at constant volume).

If the amount of substance is taken to be the number of moles in the sample (as is sometimes done in chemistry), one gets the molar heat capacity (whose SI unit is joule per kelvin per mole, J⋅K⁻¹⋅mol⁻¹).

Definition

The volumetric heat capacity is defined as

s(T)={\frac {C(T)}{V(T)}}={\frac {1}{V(T)}}\lim _{\Delta T\to 0}{\frac {\Delta Q(T)}{\Delta T}}

where $V(T)$ is the volume of the sample at temperature $T$ , and $\Delta Q(T)$ is the amount of heat energy needed to raise the temperature of the sample from $T$ to $T+\Delta T$ . This parameter is an intensive property of the substance.

Since both the heat capacity of an object and its volume may vary with temperature, in unrelated ways, the volumetric heat capacity is usually a function of temperature too. It is equal to the specific heat $c(T)$ of the substance times its density (mass per volume) $\rho (T)$ , both measured at the temperature $T$ . Its SI unit is joule per kelvin per cubic meter (J⋅K⁻¹⋅m⁻³).

This quantity is used almost exclusively for liquids and solids, since for gases it may be confused with the "specific heat capacity at constant volume", which generally has very different values. International standards now recommend that "specific heat capacity" always refer to capacity per unit of mass.^[2] Therefore, the word "volumetric" should always be used for this quantity.

History

Dulong and Petit predicted in 1818 ^{[ citation needed ]} that the product of solid substance density and specific heat capacity (ρc_p) would be constant for all solids. This amounted to a prediction that volumetric heat capacity in solids would be constant. In 1819 they found that volumetric heat capacities were not quite constant, but that the most constant quantity was the heat capacity of solids adjusted by the presumed weight of the atoms of the substance, as defined by Dalton (the Dulong–Petit law). This quantity was proportional to the heat capacity per atomic weight (or per molar mass), which suggested that it is the heat capacity per atom (not per unit of volume) which is closest to being a constant in solids.

Eventually it became clear that heat capacities per particle for all substances in all states are the same, to within a factor of two, so long as temperatures are not in the cryogenic range.

Typical values

The volumetric heat capacity of solid materials at room temperatures and above varies widely, from about 1.2 MJ⋅K⁻¹⋅m⁻³ (for example bismuth ^[3]) to 3.4 MJ⋅K⁻¹⋅m⁻³ (for example iron^[4]). This is mostly due to differences in the physical size of atoms. Atoms vary greatly in density, with the heaviest often being more dense, and thus are closer to taking up the same average volume in solids than their mass alone would predict. If all atoms were the same size, molar and volumetric heat capacity would be proportional and differ by only a single constant reflecting ratios of the atomic molar volume of materials (their atomic density). An additional factor for all types of specific heat capacities (including molar specific heats) then further reflects degrees of freedom available to the atoms composing the substance, at various temperatures.

For most liquids, the volumetric heat capacity is narrower, for example octane at 1.64 MJ⋅K⁻¹⋅m⁻³ or ethanol at 1.9. This reflects the modest loss of degrees of freedom for particles in liquids as compared with solids.

However, water has a very high volumetric heat capacity, at 4.18 MJ⋅K⁻¹⋅m⁻³, and ammonia is also fairly high: 3.3 MJ⋅K⁻¹⋅m⁻³.

For gases at room temperature, the range of volumetric heat capacities per atom (not per molecule) only varies between different gases by a small factor less than two, because every ideal gas has the same molar volume. Thus, each gas molecule occupies the same mean volume in all ideal gases, regardless of the type of gas (see kinetic theory). This fact gives each gas molecule the same effective "volume" in all ideal gases (although this volume/molecule in gases is far larger than molecules occupy on average in solids or liquids). Thus, in the limit of ideal gas behavior (which many gases approximate except at low temperatures and/or extremes of pressure) this property reduces differences in gas volumetric heat capacity to simple differences in the heat capacities of individual molecules. As noted, these differ by a factor depending on the degrees of freedom available to particles within the molecules.

Volumetric heat capacity of gases

Large complex gas molecules may have high heat capacities per mole (of molecules), but their heat capacities per mole of atoms are very similar to those of liquids and solids, again differing by less than a factor of two per mole of atoms. This factor of two represents vibrational degrees of freedom available in solids vs. gas molecules of various complexities.

In monatomic gases (like argon) at room temperature and constant volume, volumetric heat capacities are all very close to 0.5 kJ⋅K⁻¹⋅m⁻³, which is the same as the theoretical value of ⁠3/2⁠RT per kelvin per mole of gas molecules (where R is the gas constant and T is temperature). As noted, the much lower values for gas heat capacity in terms of volume as compared with solids (although more comparable per mole, see below) results mostly from the fact that gases under standard conditions consist of mostly empty space (about 99.9% of volume), which is not filled by the atomic volumes of the atoms in the gas. Since the molar volume of gases is very roughly 1000 times that of solids and liquids, this results in a factor of about 1000 loss in volumetric heat capacity for gases, as compared with liquids and solids. Monatomic gas heat capacities per atom (not per molecule) are decreased by a factor of 2 with regard to solids, due to loss of half of the potential degrees of freedom per atom for storing energy in a monatomic gas, as compared with regard to an ideal solid. There is some difference in the heat capacity of monatomic vs. polyatomic gasses, and also gas heat capacity is temperature-dependent in many ranges for polyatomic gases; these factors act to modestly (up to the discussed factor of 2) increase heat capacity per atom in polyatomic gases, as compared with monatomic gases. Volumetric heat capacities in polyatomic gases vary widely, however, since they are dependent largely on the number of atoms per molecule in the gas, which in turn determines the total number of atoms per volume in the gas.

The volumetric heat capacity is defined as having SI units of J/(m³⋅K). It can also be described in Imperial units of BTU/(ft³⋅°F).

Volumetric heat capacity of solids

Since the bulk density of a solid chemical element is strongly related to its molar mass (usually about 3R per mole, as noted above), there exists noticeable inverse correlation between a solid's density and its specific heat capacity on a per-mass basis. This is due to a very approximate tendency of atoms of most elements to be about the same size, despite much wider variations in density and atomic weight. These two factors (constancy of atomic volume and constancy of mole-specific heat capacity) result in a good correlation between the volume of any given solid chemical element and its total heat capacity. Another way of stating this, is that the volume-specific heat capacity (volumetric heat capacity) of solid elements is roughly a constant. The molar volume of solid elements is very roughly constant, and (even more reliably) so also is the molar heat capacity for most solid substances. These two factors determine the volumetric heat capacity, which as a bulk property may be striking in consistency. For example, the element uranium is a metal which has a density almost 36 times that of the metal lithium, but uranium's volumetric heat capacity is only about 20% larger than lithium's.

Since the volume-specific corollary of the Dulong–Petit specific heat capacity relationship requires that atoms of all elements take up (on average) the same volume in solids, there are many departures from it, with most of these due to variations in atomic size. For instance, arsenic, which is only 14.5% less dense than antimony, has nearly 59% more specific heat capacity on a mass basis. In other words; even though an ingot of arsenic is only about 17% larger than an antimony one of the same mass, it absorbs about 59% more heat for a given temperature rise. The heat capacity ratios of the two substances closely follows the ratios of their molar volumes (the ratios of numbers of atoms in the same volume of each substance); the departure from the correlation to simple volumes in this case is due to lighter arsenic atoms being significantly more closely packed than antimony atoms, instead of similar size. In other words, similar-sized atoms would cause a mole of arsenic to be 63% larger than a mole of antimony, with a correspondingly lower density, allowing its volume to more closely mirror its heat capacity behavior.

Volumetric heat capacity of liquids

The volumetric heat capacity of liquids could be measured from the thermal conductivity and thermal diffusivity correlation. The volumetric heat capacity of liquids could be directly obtained during thermal conductivity analysis using thermal conductivity analyzers that use techniques like the transient plane source method.^[5]

Constant volume and constant pressure

For gases it is necessary to distinguish between volumetric heat capacity at constant volume and volumetric heat capacity at constant pressure, which is always larger due to the pressure–volume work done as a gas expands during heating at constant pressure (thus absorbing heat which is converted to work). The distinctions between constant-volume and constant-pressure heat capacities are also made in various types of specific heat capacity (the latter meaning either mass-specific or mole-specific heat capacity).

Thermal inertia

Thermal inertia is a term commonly used to describe the observed delays in a body's temperature response during heat transfers. The phenomenon exists because of a body's ability to both store and transport heat relative to its environment. Larger values of volumetric heat capacity, as may occur in association with thermal effusivity, typically yield slower temperature responses.

Related Research Articles

In thermodynamics, the specific heat capacity of a substance is the amount of heat that must be added to one unit of mass of the substance in order to cause an increase of one unit in temperature. It is also referred to as Massic heat capacity or as the Specific heat. More formally it is the heat capacity of a sample of the substance divided by the mass of the sample. The SI unit of specific heat capacity is joule per kelvin per kilogram, J⋅kg⁻¹⋅K⁻¹. For example, the heat required to raise the temperature of 1 kg of water by 1 K is 4184 joules, so the specific heat capacity of water is 4184 J⋅kg⁻¹⋅K⁻¹.

The Avogadro constant, commonly denoted $N A$ or $L$ , is an SI defining constant with an exact value of 6.02214076×10²³ mol⁻¹ (reciprocal moles). It is this defined number of constituent particles (usually molecules, atoms, ions, or ion pairs—in general, entities) per mole (SI unit) and used as a normalization factor in relating the amount of substance, n(X), in a sample of a substance X to the corresponding number of entities, N(X): n(X) = N(X)(1/N_A), an aggregate of N(X) reciprocal Avogadro constants. By setting N(X) = 1, a reciprocal Avogadro constant is seen to be equal to one entity, which means that n(X) is more easily interpreted as an aggregate of N(X) entities. In the SI dimensional analysis of measurement units, the dimension of the Avogadro constant is the reciprocal of amount of substance, denoted N⁻¹. The Avogadro number, sometimes denoted $N 0$ , is the numeric value of the Avogadro constant (i.e., without a unit), namely the dimensionless number 6.02214076×10²³; the value chosen based on the number of atoms in 12 grams of carbon-12 in alignment with the historical definition of a mole. The constant is named after the Italian physicist and chemist Amedeo Avogadro (1776–1856).

Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.

<span class="mw-page-title-main">Gas constant</span> Physical constant equivalent to the Boltzmann constant, but in different units

The molar gas constant is denoted by the symbol $R$ or $R$ . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per amount of substance, rather than energy per temperature increment per particle. The constant is also a combination of the constants from Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. It is a physical constant that is featured in many fundamental equations in the physical sciences, such as the ideal gas law, the Arrhenius equation, and the Nernst equation.

An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics. The requirement of zero interaction can often be relaxed if, for example, the interaction is perfectly elastic or regarded as point-like collisions.

The van der Waals radius, r_w, of an atom is the radius of an imaginary hard sphere representing the distance of closest approach for another atom. It is named after Johannes Diderik van der Waals, winner of the 1910 Nobel Prize in Physics, as he was the first to recognise that atoms were not simply points and to demonstrate the physical consequences of their size through the van der Waals equation of state.

In chemistry and related fields, the molar volume, symbol V_m, or $of a substance is the ratio of the volume (V) occupied by a substance to the amount of substance (n), usually at a given temperature and pressure. It is also equal to the molar mass (M) divided by the mass density (ρ):$

The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. More simply, the speed of sound is how fast vibrations travel. At 20 °C (68 °F), the speed of sound in air, is about 343 m/s, or 1 km in 2.91 s or one mile in 4.69 s. It depends strongly on temperature as well as the medium through which a sound wave is propagating.

In chemistry, the amount of substance (symbol $n$ ) in a given sample of matter is defined as a ratio ( $n = N / N A$ ) between the number of elementary entities ( $N$ ) and the Avogadro constant ( $N A$ ). The entities are usually molecules, atoms, ions, or ion pairs of a specified kind. The particular substance sampled may be specified using a subscript, e.g., the amount of sodium chloride (NaCl) would be denoted as $n NaCl$ . The unit of amount of substance in the International System of Units is the mole (symbol: mol), a base unit. Since 2019, the value of the Avogadro constant $N A$ is defined to be exactly 6.02214076×10²³ mol⁻¹. Sometimes, the amount of substance is referred to as the chemical amount or, informally, as the "number of moles" in a given sample of matter.

The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. Alternatively, it is the heat capacity of a sample of the substance divided by the amount of substance of the sample; or also the specific heat capacity of the substance times its molar mass. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K⁻¹⋅mol⁻¹.

In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor and is denoted by $γ$ (gamma) for an ideal gas or $κ$ (kappa), the isentropic exponent for a real gas. The symbol $γ$ is used by aerospace and chemical engineers. $where C is the heat capacity, the molar heat capacity, and c the specific heat capacity of a gas. The suffixes P and V refer to constant-pressure and constant-volume conditions respectively.$

In thermodynamics, the specific volume of a substance is the quotient of the substance's volume to its mass :

The number density is an intensive quantity used to describe the degree of concentration of countable objects in physical space: three-dimensional volumetric number density, two-dimensional areal number density, or one-dimensional linear number density. Population density is an example of areal number density. The term number concentration is sometimes used in chemistry for the same quantity, particularly when comparing with other concentrations.

This glossary of chemistry terms is a list of terms and definitions relevant to chemistry, including chemical laws, diagrams and formulae, laboratory tools, glassware, and equipment. Chemistry is a physical science concerned with the composition, structure, and properties of matter, as well as the changes it undergoes during chemical reactions; it features an extensive vocabulary and a significant amount of jargon.

<span class="mw-page-title-main">Gas</span> State of matter

Gas is one of the four fundamental states of matter. The others are solid, liquid, and plasma. A pure gas may be made up of individual atoms, elemental molecules made from one type of atom, or compound molecules made from a variety of atoms. A gas mixture, such as air, contains a variety of pure gases. What distinguishes gases from liquids and solids is the vast separation of the individual gas particles. This separation usually makes a colorless gas invisible to the human observer.

In thermodynamics, the volume of a system is an important extensive parameter for describing its thermodynamic state. The specific volume, an intensive property, is the system's volume per unit mass. Volume is a function of state and is interdependent with other thermodynamic properties such as pressure and temperature. For example, volume is related to the pressure and temperature of an ideal gas by the ideal gas law. The physical region covered by a system may or may not coincide with a control volume used to analyze the system.

In thermodynamics, the enthalpy of fusion of a substance, also known as (latent) heat of fusion, is the change in its enthalpy resulting from providing energy, typically heat, to a specific quantity of the substance to change its state from a solid to a liquid, at constant pressure.

The Dulong–Petit law, a thermodynamic law proposed by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states that the classical expression for the molar specific heat capacity of certain chemical elements is constant for temperatures far from the absolute zero.

In physics and chemistry, "monatomic" is a combination of the words "mono" and "atomic", and means "single atom". It is usually applied to gases: a monatomic gas is a gas in which atoms are not bound to each other. Examples at standard conditions of temperature and pressure include all the noble gases, though all chemical elements will be monatomic in the gas phase at sufficiently high temperature. The thermodynamic behavior of a monatomic gas is much simpler when compared to polyatomic gases because it is free of any rotational or vibrational energy.

The table of specific heat capacities gives the volumetric heat capacity as well as the specific heat capacity of some substances and engineering materials, and the molar heat capacity.

References

↑ "U.S. Army Corps of Engineers Technical Manual: Arctic and Subarctic Construction: Calculation Methods for Determination of Depths of Freeze and Thaw in Soils, TM 5-852-6/AFR 88-19, Volume 6, 1988, Equation 2-1" (PDF). Archived from the original (PDF) on 2016-03-04. Retrieved 2015-05-19.
↑ International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN 92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16
↑ Based on values in this table and density.
↑ Based on NIST data and density.
↑ Sreekumar, Sreehari.; Ganguly, Abhijit.; Khalil, Sameh.; Chakrabarti, Supriya.; Hewitt, Neil.; Mondol, Jayanta.; Shah, Nikhilkumar. (2023). "Thermo-optical characterization of novel MXene/Carbon-dot hybrid nanofluid for heat transfer applications". Journal of Cleaner Production. 434 (29): 140395. doi: 10.1016/j.jclepro.2023.140395 .

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[1] "U.S. Army Corps of Engineers Technical Manual: Arctic and Subarctic Construction: Calculation Methods for Determination of Depths of Freeze and Thaw in Soils, TM 5-852-6/AFR 88-19, Volume 6, 1988, Equation 2-1" (PDF). Archived from the original (PDF) on 2016-03-04. Retrieved 2015-05-19.

[2] International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN 92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16

[3] Based on values in this table and density.

[4] Based on NIST data and density.

[5] Sreekumar, Sreehari.; Ganguly, Abhijit.; Khalil, Sameh.; Chakrabarti, Supriya.; Hewitt, Neil.; Mondol, Jayanta.; Shah, Nikhilkumar. (2023). "Thermo-optical characterization of novel MXene/Carbon-dot hybrid nanofluid for heat transfer applications". Journal of Cleaner Production. 434 (29): 140395. doi: 10.1016/j.jclepro.2023.140395 .

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